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Full text of "Text Book Of Mechanical Engineering"

Appendix VI.

1087

Change of Pole. In Fig. 961 are shewn six forces exactly
balancing, as proven from the vector figure. Let the link polygon
be drawn in the first place for a pole 0, and let it be desired to
re-draw it for a new pole Or This has been done point by point
in the usual manner, arriving at the dotted polygon, but could
have been effected more simply by the use of a  simple property
connected with two poles of the same vector figure. Join O Ox
producing both ways. Next take any line of one link polygon
and the corresponding line on the other link polygon; produce
these till they meet, obtaining, say, the point <?3. Doing this for
all the corresponding sets of lines of the polygons, we have the
additional points ^ <?0 ^. It will now be found that these points
are in one straight line, that is itself parallel to the lines O Or

For proof:                    ^                               ^

01 = oO-fOi        also, 01 = oC^-hC^i

But because 01 balances 10,
oO-r-Oi balances C^o-}- iOl
and oO-f-Oi+OjO-J-iC^ = zero
But OjO+oO = O O1
and Oi-f iOx =0^0

The force O Oj is therefore the resultant of the corresponding
link pairs in either link polygon. Its reciprocal el <?0 must join
the meets of the link pairs, and must be parallel to O Or A
similar treatment of other pairs will prove the line <?3 to e also
parallel to 0 Ox.

Use of Link Polygon.As an example take a semicircular
roof, of which half is shewn at A B c, Fig. 962. Let the wind
blow from right to left, and we require to know the direction and
magnitude of the resultant of all the pressure that comes upon
c B, in order to carry out the construction given at p. 470, and
thus arrive at the stresses on the members. Divide the arc B c
into any convenient number of equal parts at r>, E, F, G, H, j,
K, L, in this case nine. Assume the pressure intensity uniform
over each division, though varying from one division to anpther,
and let each piece of area acted upon be H x br, where K is the
height and b' the breadth between the bays, both in feet. If the
wind intensity be w, the normal intensity n will vary from w at B
to zero at c, and its value at any point can be found from the